How to find the length of the diagonal of a kite - ACT Math

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Question

A kite has two perpendicular interior diagonals. One diagonal is twice the length of the other diagonal. The total area of the kite is $196\textup{ units}^{2}$ . Find the length of each interior diagonal.

Answer

To solve this problem, apply the formula for finding the area of a kite:

$Area=\frac{diagonalA\times diagonalB}{2}$

However, in this problem the question only provides information regarding the exact area. The lengths of the diagonals are represented as a ratio, where

Therefore, it is necessary to plug the provided information into the area formula. Diagonal is represented by and diagonal .

The solution is:

$196=\frac{x\times 2x}{2}$

$196\times2=x\times 2x$

$x^2=\frac{392}{2}=196$

$x=\sqrt{196}=14$

Thus, if , then diagonal must equal

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $60\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Answer

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$60=\frac{8\times diagonal B}{2}$

$60\times2=8\times diagonalB$

$diagonal B=\frac{120}{8}=15$

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Question

Using the kite shown above, find the length of the red (vertical) diagonal.

Answer

In order to solve this problem, first observe that the red diagonal line divides the kite into two triangles that each have side lengths of and Notice, the hypotenuse of the interior triangle is the red diagonal. Therefore, use the Pythagorean theorem: , where the length of the red diagonal.

The solution is:

$c=\sqrt{289}=\sqrt{17\times 17}=17$

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $45\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Answer

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$45=\frac{18\times diagonal B}{2}$

$45\times2=18\times diagonalB$

$diagonal B=\frac{90}{18}=5$

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $6\textup{,}250\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Answer

First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$6,250=\frac{250\times diagonal B}{2}$

$6250\times2=250\times diagonalB$

$diagonal B=\frac{12,500}{250}=50$

Therefore, the sum of the two diagonals is:

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $28\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Answer

You must find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$28=\frac{4\times diagonal B}{2}$

$28\times2=4\times diagonalB$

$diagonal B=\frac{56}{4}=14$

Therefore, the sum of the two diagonals is:

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Question

The area of the kite shown above is $125\textup{ units}^{2}$ and the red diagonal has a length of . Find the length of the black (horizontal) diagonal.

Answer

To find the length of the black diagonal apply the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$125=\frac{25\times diagonal B}{2}$

$125\times2=25\times diagonalB$

$diagonal B=\frac{250}{25}=10$

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of $22\textup{mm}$ and the area of the kite is $297\textup{mm}^{2}$ . Find the length of the other interior diagonal.

Answer

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$297=\frac{22\times diagonal B}{2}$

$297\times2=22\times diagonalB$

$diagonal B=\frac{594}{22}=27$

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $1\textup{,}800\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Answer

You must find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$1800=\frac{40\times diagonal B}{2}$

$1800\times2=40\times diagonalB$

$diagonal B=\frac{3,600}{40}=90$

Therefore, the sum of the two diagonals is:

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $54\textup{ units}^{2}$ . Find the length of the other interior diagonal.

Answer

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$54=\frac{9\times diagonal B}{2}$

$54\times2=9\times diagonalB$

$diagonal B=\frac{108}{9}=12$

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of and the area of the kite is $51\textup{ units}^{2}$ . Find the sum of the two perpendicular interior diagonals.

Answer

First find the length of the missing diagonal before you can find the sum of the two perpendicular diagonals.

To find the missing diagonal, apply the area formula:

This question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

$51=\frac{17\times diagonal B}{2}$

$51\times2=17\times diagonalB$

$diagonal B=\frac{102}{17}=6$

Therefore, the sum of the two diagonals is:

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Question

A kite has two perpendicular interior diagonals. One diagonal has a measurement of $38\textup{cm}$ and the area of the kite is $855\textup{cm}^{2}$ . Find the length of the other interior diagonal.

Answer

This problem can be solved by applying the area formula:

Since this question provides the area of the kite and length of one diagonal, plug that information into the equation to solve for the missing diagonal.

Thus the solution is:

$855=\frac{38\times diagonal B}{2}$

$855\times2=38\times diagonalB$

$diagonal B=\frac{1,710}{38}=45$

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Question

The long diagonal of a kite measures inches, and cuts the shorter diagonal into two pieces. If one of those pieces measures inches, what is the length in inches of the short diagonal?

Answer

The long diagonal of a kite always bisects the short diagonal. Therefore, if one side of the bisected diagonal is inches, the entire diagonal is inches. It does not matter how long the long diagonal is.

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